Pentagon integrals to arbitrary order in the dimensional regulator

Abstract

We analytically calculate one-loop five-point Master Integrals, pentagon integrals, with up to one off-shell leg to arbitrary order in the dimensional regulator in d = 4−2𝜖 space-time dimensions. A pure basis of Master Integrals is constructed for the pentagon family with one off-shell leg, satisfying a single-variable canonical differential equation in the Simplified Differential Equations approach. The relevant boundary terms are given in closed form, including a hypergeometric function which can be expanded to arbitrary order in the dimensional regulator using the Mathematica package HypExp. Thus one can obtain solutions of the canonical differential equation in terms of Goncharov Polylogartihms of arbitrary transcendental weight. As a special limit of the one-mass pentagon family, we obtain a fully analytic result for the massless pentagon family in terms of pure and universally transcendental functions. For both families we provide explicit solutions in terms of Goncharov Polylogartihms up to weight four.

One-loop Feynman integrals for 2 → 3 scattering involving many scales including internal masses

“…The structure of this alphabet is similar in terms of its complexity with the alphabet for the one-mass pentagon studied in [35]. The difference of course is the presence of an additional square root of the underline kinematic variables S ij in the present case.…”

“…Equation (3.14) allows us to fix all the necessary boundary terms without the need of any further computation for the families C, E, H while for family G a few regions had to be computed. Similarly to [35], the resulting boundary terms for all of the four families considered in this subsection are in closed form, including some 2 F 1 Hypergeometric functions which can be easily expanded to arbitrary powers of the dimensional regulator using HypExp [47]. Therefore we are able to trivially obtain solutions of (3.9) for the families C, E, G, H in terms of Goncharov polylogarithms of arbitrary weight.…”

“…On a more general note, whenever the Simplified Differential Equations approach has been applied in conjunction with a pure basis [31,35,38], we have observed a reduction in the number of letters that appear when compared with the alphabets that arise through the usual method of differential equations, i.e. when one differentiates with respect to all kinematic invariants.…”

“…The basic rule for choosing which five-point family to consider is to have the one-mass result [35] as a starting point and add masses, with the condition that their simplified differential equations in canonical form have alphabets rational in x, when one uses parametrization (2.3). If the resulting alphabet is not rational in x, then calculating the integral family through the x → 1 limit of another is considered.…”

“…W + 2 jets production. For the computation of the relevant scattering amplitudes, one-loop five-point Feynman integrals with one off-shell leg also have to be known up to transcendental weight four [35]. These results were recently used for the calculation of two-loop QCD corrections to W bb production [36].…”